Deconvolutional determination of the nonlinearity in a semilinear wave equation
Nicholas Hu, Rowan Killip, Monica Visan
本文证明了在三维空间中，具有五次型非线性的半线性波动方程的散射行为能唯一确定同时依赖时空的非线性。
我们证明，在三维空间中，具有五次型非线性的半线性波动方程的散射行为能唯一确定该非线性。该非线性允许同时依赖于空间和时间。
We demonstrate that in three space dimensions, the scattering behaviour of semilinear wave equations with quintic-type nonlinearities uniquely determines the nonlinearity. The nonlinearity is permitted to depend on both space and time.
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We consider a semilinear elliptic equation in a bounded domain with zero boundary conditions. The nonlinearity is discontinuous and monotone, but it is not a Carathéodory's function. The existence theorem has been proved.
Unique inversion of smooth nonlinearity in semilinear wave equations on Lorentzian manifolds
研究洛伦兹流形上半线性波动方程光滑非线性项的恢复，核心方法是结合高阶线性化与扭曲平面波、高斯光束，利用多波相互作用恢复泰勒系数。
We study two-dimensional semilinear strongly damped wave equation with mixed nonlinearity $|u|^p+|u_t|^q$ in an exterior domain, where $p,q>1$. Assuming the smallness of initial data in exponentially weighted spaces and some conditions on powers of nonlinearity, we prove global (in time) existence of small data energy solution with suitable higher regularity by using a weighted energy method.
We obtain a lower bound for the period of periodic solutions of semilinear evolution equations for the full range of nonlinear terms for which standard local existence theory applies. This lower bound depends on the Lipschitz constant of the nonlinear term as an operator acting on the domain of a fractional power of the linear operator into the base space.
In this paper, we study the blow-up of solutions to the semilinear Moore-Gibson-Thompson (MGT) equation with nonlinearity of derivative type $|u_t|^p$ in the conservative case. We apply an iteration method in order to study both the subcritical case and the critical case. Hence, we obtain a blow-up result for the semilinear MGT equation (under suitable assumptions for initial data) when the exponent $p$ for the nonlinear term satisfies $1<p\leqslant (n+1)/(n-1)$ for $n\geqslant2$ and $p>1$ for $n=1$. In particular, we find the same blow-up range for $p$ as in the corresponding semilinear wave …
In this paper, we study the blow-up of solutions for semilinear wave equations with scale-invariant dissipation and mass in the case in which the model is somehow 'wave-like'. A Strauss type critical exponent is determined as the upper bound for the exponent in the nonlinearity in the main theorems. Two blow-up results are obtained for the sub-critical case and for the critical case, respectively. In both cases, an upper bound lifespan estimate is given.
Mixed local and nonlocal semilinear elliptic equation with strongly singular and critical Choquard nonlinearity
本文研究混合局部与非局部椭圆方程，结合奇异与临界非线性，通过变分方法证明正解的存在性与多重性。
In this note, we prove a blow-up result for the semilinear damped wave equation in a compact Lie group with power nonlinearity $|u|^p$ for any $p>1$, under suitable integral sign assumptions for the initial data, by using an iteration argument. A byproduct of this method is the upper bound estimate for the lifespan of a local in time solution. As a preliminary result, a local (in time) existence result is proved in the energy space via Fourier analysis on compact Lie groups.
We consider positive solutions $u$ of the semilinear biharmonic equation $Δ^2 u = |x|^{-\frac{n+4}{2}} g(|x|^\frac{n-4}{2} u)$ in $\mathbb R^n \setminus \{0\}$ with non-removable singularities at the origin. Under natural assumptions on the nonlinearity $g$, we show that $|x|^\frac{n-4}{2} u$ is a periodic function of $\ln |x|$ and we classify all such solutions.
Given $k\in\mathbb N$, we define a class of continuous piecewise functions $f$ having abrupt but controlled magnitude changes so that the problem $$Δu +f(u)=0,\quad x\in \mathbb R^N, N> 2, $$ has at least $k$ radially symmetric ground state solutions.
On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlinearity
该研究利用变分法与逼近方法，考察兼具局部与非局部 Laplace 算子的奇异半线性椭圆方程，证明参数依赖情形下弱解存在及扰动情形下多解存在。
In this manuscript, in the $L^1$ scaling critical case, a lifespan estimate of solutions to the Cauchy problem for non-gauge invariant semilinear semirelativistic equations is considered. The lifespan estimate is given by the modified test function method with a fractional Laplace operator. The main obstacle to obtaining the lifespan estimate is the non-locality of the fractional Laplace operator. To treat the non-locality, special test functions are introduced.
Existence of solutions to the semilinear damped wave equation with non-$L^2$ slowly decaying data : polynomial nonlinearity case
研究半线性阻尼波方程的初值问题，针对非L²缓慢衰减初始数据，建立局部和全局解的存在性，方法基于线性解的L^p-L^q估计和合适的小波空间中的分数Leibniz规则。
On a bounded smooth domain we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to the boundary of the domain. We derive global a priori bounds of the Keller-Osserman type. Using a Phragmen-Lindelof alternative for generalized sub and super-harmonic functions we discuss existence, nonexistence and uniqueness of so-called large solutions, i.e., solutions which tend to infinity at the boundary. The approach develops the one used by the same authors for a problem with a power nonlinearity instead of the exponential n…
A blow-up result for the semilinear Euler-Poisson-Darboux-Tricomi equation with critical power nonlinearity
本文针对具有多项式增长传播速度的广义半线性Euler-Poisson-Darboux方程，在临界幂情形下证明了爆破结果，并给出了与经典半线性波动方程一致的生命跨度上界估计。
We consider in this paper blow-up solutions of the semilinear wave equation in one space dimension, with an exponential source term. Assuming that initial data are in $H^{1}_{loc}\times L^2_{loc}$ or some times in $ W^{1,\infty}\times L^{\infty}$, we derive the blow-up rate near a non-characteristic point in the smaller space, and give some bounds near other points. Our result generalize those proved by Godin under high regularity assumptions on initial data.
In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integral Transform approach. As upper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent which depends both on the space dimension and on the Lorentzian metric in the generalized Einstein-de Sitter spacetime.
Rigidity for a semilinear Neumann problem with exponential nonlinearity in the large diffusion limit
研究在大扩散极限下，指数非线性半线性Neumann问题的解的刚性性质，证明存在阈值ε，使得ε>阈值时所有经典解必须为常数，回答了Calanchi等人提出的猜想。
The lifespan of solutions of semilinear wave equation with weighted nonlinearity
研究带加权非线性项的半线性波动方程在小初值下解的生命跨度，分析权重对解寿命的影响。
In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power nonlinearity. We prove that the critical exponent is the Fujita exponent $p_{\mathrm{Fuj}}(\mathscr{Q}) = 1+2 / \mathscr{Q}$, where $\mathscr{Q}$ is the homogeneous dimension of the Heisenberg group. On the one hand, we will prove the global existence of small data solutions for $p >p_{\mathrm{Fuj}}(\mathscr{Q})$ in an exponential weighted energy space. On the other hand, a blow-up result for $1 < p \leq p_{\mathrm{Fuj}}(\mathscr{Q})$ under certain integral sign assumptions …
On the the critical exponent for the semilinear Euler-Poisson-Darboux-Tricomi equation with power nonlinearity
本文推导了带有时变系数阻尼项和质量项的半线性广义特里科米方程的爆破结果，研究临界衰减率下系数的阈值性质对临界指数的影响，揭示藤田型指数与施特劳斯型指数的竞争关系。
We consider the semilinear heat equation $u_t=Δu+|u|^{p-1}u-|u|^{q-1}u$ in $\mathbb{R}^n\times(0,T)$, where $n=5$, $p=\frac{n+2}{n-2}$ and $q\in(0,1)$. By the presence of $-|u|^{q-1}u$, this equation has a finite time extinction property. We show the existence of a new type of blowup solutions by using this property. In fact, we obtain such blowup solutions by connecting a specific blowup solution of $u_t=Δu+|u|^{p-1}u$ and a specific solution of $u_t=Δu-|u|^{q-1}u$, and by adding correction terms.
An exact traveling waves solution for a special class of semilinear equations
本文针对含特定非线性项的半线性抛物方程，构造了精确行波解，该非线性项源于分支布朗运动投票模型，此前相关结果均为其特例。
Semilinear elliptic problems on the half space with a supercritical nonlinearity
本文研究半空间上具Sobolev超临界非线性项和非齐次Dirichlet边界条件的标量场方程正解，在边界数据可积且非线性项满足Joseph--Lundgren次临界条件下，完整分类了正解的存在性，并通过分歧理论证明了多重正解的存在。
We study the inverse problem of recovery a non-linearity $f(x,u)$, which is compactly supported in $x$, in the semilinear wave equation $u_{tt}-Δu+ f(x,u)=0$. We probe the medium with either complex or real-valued harmonic waves of wavelength $\sim h$ and amplitude $\sim 1$. They propagate in a regime where the non-linearity affects the subprincipal but not the principal term, except for the zeroth harmonics. We measure the transmitted wave when it exits $\text{supp}_x f$. We show that one can recover $f(x,u)$ when it is an odd function of $u$, and we can recover $α(x)$ when $f(x,u)=α(x)u^{2m}…
We show that singularities form after the interaction of three transversal semilinear conormal waves. Our results hold for space dimensions two and higher, and for arbitrary smooth nonlinearity. The case of two space dimensions in which the nonlinearity is a polynomial was studied by the author and Yiran Wang.
Inverse problems for semilinear elliptic PDE with a general nonlinearity $a(x,u)$
本文研究通过边界测量恢复椭圆方程中非线性项的问题，改进了基于一阶和高阶线性化的方法，证明一般非线性项在固定解附近唯一确定。
Well-posedness and linearization for a semilinear wave equation with spatially growing nonlinearity
本文研究了具有空间增长非线性的去聚焦半线性波方程的初值问题，通过联合运用 Strauss 不等式和 Strichartz 估计，在径向框架内克服了无穷远处非线性增长带来的挑战，建立了能量空间中的全局适定性并推导了能量有界解的线性化。
We study the existence of solutions to the fractional elliptic equation (E1) $(-Δ)^αu+εg(|\nabla u|)=ν$ in a bounded regular domain $Ω$ of $\R^N (N\ge2)$, subject to the condition (E2) $u=0$ in $Ω^c$, where $ε=1$ or $-1$, $(-Δ)^α$ denotes the fractional Laplacian with $α\in(1/2,1)$, $ν$ is a Radon measure and $g:\R_+\mapsto\R_+$ is a continuous function. We prove the existence of weak solutions for problem (E1)-(E2) when $g$ is subcritical. Furthermore, the asymptotic behavior and uniqueness of solutions are described when $ν$ is Dirac mass, $g(s)=s^p$, $p\geq 1$ and $ε=1$.
Multiple solutions to a semilinear elliptic equation with a sharp change of sign in the nonlinearity
本文研究非线性项系数在有界开集内外取相反符号的半线性椭圆方程，利用Lyapunov-Schmidt约化方法证明了次临界情形下集中正解与节点解的存在性及正解的多重性，并在临界情形下证明了非平凡拓扑区域中爆破解的存在性。
We consider semilinear wave equations with small initial data in two space dimensions. For a class of wave equations with cubic nonlinearity, we show the global existence of small amplitude solutions, and give an asymptotic description of the solution as $t \to \infty$ uniformly in $x \in {\mathbb R}^2$. In particular, our result implies the decay of the energy when the nonlinearity is dissipative.
In this paper we study multiplicity and qualitative behavior of solutions for semilinear elliptic problems with neumann boundary condition and asymptotically linear smooth nonlinearity. We provide sufficient conditions on the number of eigenvalues the derivative of the nonlinearity crosses to guarantee existence of at least five nontrivial solutions. The techniques we use are a combination of minimization, Leray-Schauder degree, Morse Theory and Reduction method a la Castro-Lazer.
We study the semilinear elliptic equation \begin{equation*} -Δu=u^α|\log u|^β\quad\text{in }B_1\setminus\{0\}, \end{equation*} where $B_1\subset\mathbb{R}^n$ with $n\geq 3$, $\frac{n}{n-2} < α< \frac{n+2}{n-2}$ and $-\infty<β<\infty$. Our main result establishes that nonnegative solution $u\in C^2(B_1\setminus\{0\})$ of the above equation either has a removable singularity at the origin or behaves like \begin{equation*} u(x) = A(1+o(1)) |x|^{-\frac{2}{α-1}} \left(\log \frac{1}{|x|}\right)^{-\fracβ{α-1}}\quad\text{as } x\rightarrow 0, \end{equation*} with \begin{equation*} A=\left[\left(\frac{2…
In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. I…
In this work, the Cauchy problem for the semilinear Moore-Gibson-Thompson (MGT) equation with power nonlinearity $|u|^p$ on the right-hand side is studied. Applying $L^2-L^2$ estimates and a fixed point theorem, we obtain local (in time) existence of solutions to the semilinear MGT equation. Then, the blow-up of local in time solutions is proved by using an iteration method, under certain sign assumption for initial data, and providing that the exponent of the power of the nonlinearity fulfills $1 < p \leqslant p_{\mathrm{Str}}(n)$ for $n \geqslant2$ and $p>1$ for $n=1$. Here the Strauss expon…
We characterize the asymptotic speed of propagation of almost planar solutions to a semilinear viscous parabolic equation, with periodic nonlinearity.
On the threshold nature of the Dini continuity for a Glassey derivative-type nonlinearity in a critical semilinear wave equation
本文研究含临界 Glassey 指数导数型非线性项的半线性波动方程，利用 Zhou 特征线方法与加权 $L^{\infty}$ 估计，证明 Dini 连续性是决定有限时间爆破与小数据全局存在的正则性阈值。
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
Existence of solutions of semilinear wave equations with time-dependent propagation speed and time derivative nonlinearity
本文研究含时变传播速度和时间导数非线性项的半线性波动方程，给出传播速度的可积性条件，证明膨胀宇宙中解的全局存在性与寿命下界，同时证明收缩宇宙中解的有限时间爆破并给出寿命上界。
In the present paper, we investigate the blow-up dynamics for local solutions to the semilinear generalized Tricomi equation with combined nonlinearity. As a result, we enlarge the blow-up region in comparison to the ones for the corresponding semilinear models with either power nonlinearity or nonlinearity of derivative type. Our approach is based on an iteration argument to establish lower bound estimates for the space average of local solutions. Finally, we obtain upper bound estimates for the lifespan of local solutions as byproduct of our iteration argument.
Stable blowup profile for a semilinear heat equation with spatially inhomogeneous nonlinearity
研究了带有额外的发散Hénon型非线性的聚焦半线性热方程，分析了其在小非径向扰动下的稳定性，并证明了在c接近c*时的稳定爆破。
Classification of bifurcation structure for semilinear elliptic equations in a ball
本文研究单位球内Gelfand问题的分岔结构，发现分岔曲线转折点数量不仅取决于非线性项增长速率与Joseph-Lundgren临界值的比较，并给出决定分岔结构的一般准则与分类。
We demonstrate the large deviation property for the mild solutions of stochastic evolution equations with monotone nonlinearity and multiplica- tive noise. This is achieved using the recently developed weak convergence method, in studying the large deviation principle. An It^o-type inequality is a main tool in the proofs. We also give two examples to illustrate the applications of the theorems.
We consider a possibly anisotropic integro-differential semilinear equation, run by a nondecreasing and nontrivial nonlinearity. We prove that if the solution grows at infinity less than the order of the operator, then it must be constant.
Consider nonlinear wave equations in the spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes. We show blow-up in finite time of solutions and upper bounds of the lifespan of blow-up solutions to give the FLRW spacetime version of Glassey's conjecture for the time derivative nonlinearity. We also show blow-up results for the space time derivative nonlinearity.
Semilinear degenerate elliptic equation in the presence of singular nonlinearity
本文研究了在光滑有界区域上，具有奇异非线性的拟线性退化椭圆方程解的存在性与正则性，其中涉及的算子为 Grushin 算子。
Sharp lifespan estimates for semilinear fractional evolution equations with critical nonlinearity
本文研究由分数阶Laplace算子驱动的具临界非线性的半线性波动与二阶σ-演化方程，通过Dini条件推导整体存在性与爆破结果，并引入新测试函数获得了局部解寿命的精确上下界估计。
We investigate inverse boundary problems associated with a time-dependent semilinear hyperbolic equation, where both nonlinearity and sources (including initial displacement and initial velocity) are unknown. We establish in several generic scenarios that one can uniquely determine the nonlinearity or the sources by using passive or active boundary observations. In order to exploit the nonlinearity and the sources simultaneously, we develop a new technique, which combines the observability for linear wave equations and a Runge approximation with higher order linearization for the semilinear hy…
We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: $-Δu=λu^p - u^q$ in $\mathbb{R}^{N+1}$, where $ 0< q < p \leq 1$, $λ\in \mathbb{R}$. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter $λ$, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the …
We study the system $-Δ\mathbf{u}=| \mathbf{u}|^{α-1} \mathbf{u}$ with $1<α\leq\frac{n+2}{n-2}$, where $ \mathbf{u}=(u_1,\dots,u_m)$, $m\geq 1$, is a $C^2$ nonnegative function that develops an isolated singularity in a domain of $\mathbb{R}^n$, $n\geq 3$. Due to the multiplicity of the components of $ \mathbf{u}$, we observe a new Pohozaev invariant other than the usual one in the scalar case, and also a new class of singular solutions provided that the new invariant is nontrivial. Aligned with the classical theory of the scalar equation, we classify the solutions on the whole space as well a…
We are concerned with the properties of weak solutions of the stationary Schrödinger equation $-Δu + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where $V$ is Hölder continuous and $\inf V>0$. Assuming $f$ to be continuous and bounded near $0$ by a power function with exponent larger than $1$ we provide precise decay estimates at infinity for solutions in terms of Green's function of the Schrödinger operator. In some cases this improves known theorems on the decay of solutions. If $f$ is also real analytic on $(0,\infty)$ we obtain that the set of positive solutions is loca…
Singular solutions and bifurcation diagram of semilinear elliptic equations with general nonlinearity in two dimensions
针对二维一般指数型非线性半线性椭圆方程，放宽奇异解存在条件并给出其渐近形式，同时在非线性项非解析时证明正则解分支曲线围绕某点无穷多次振荡。
We show that the knowledge of the Dirichlet-to-Neumann map on an arbitrary open portion of the boundary of a domain in $\mathbb{R}^n$, $n\ge 2$, for a class of semilinear elliptic equations, determines the nonlinearity uniquely.
For any subcritical index of regularity $s>3/2$, we prove the almost global well posedness for the 2-dimensional semilinear wave equation with the cubic nonlinearity in the derivatives, when the initial data are small in the Sobolev space $H^s\times H^{s-1}$ with certain angular regularity. The main new ingredient in the proof is an endpoint version of the generalized Strichartz estimates in the space $L^2_t L_{|x|}^\infty L^2_θ([0,T]\times \R^2)$. In the last section, we also consider the general semilinear wave equations with the spatial dimension $n\ge 2$ and the order of nonlinearity $p\ge…
In this paper we consider the critical exponent problem for the semilinear wave equation with space-time dependent damping. When the damping is effective, it is expected that the critical exponent agrees with that of only space dependent coefficient case. We shall prove that there exists a unique global solution for small data if the power of nonlinearity is larger than the expected exponent. Moreover, we do not assume that the data are compactly supported. However, it is still open whether there exists a blow-up solution if the power of nonlinearity is smaller than the expected exponent.
Let $Ω\subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $δ$ be the distance to $\partial Ω$. We study positive solutions of equation (E) $-L_μu+ g(|\nabla u|) = 0$ in $Ω$ where $L_μ=Δ+ \fracμ{δ^2} $, $μ\in (0,\frac{1}{4}]$ and $g$ is a continuous, nondecreasing function on ${\mathbb R}_+$. We prove that if $g$ satisfies a singular integral condition then there exists a unique solution of (E) with a prescribed boundary datum $ν$. When $g(t)=t^q$ with $q \in (1,2)$, we show that equation (E) admits a critical exponent $q_μ$ (depending only on $N$ and $μ$). In the subcritical case…
We prove that in three space dimensions a nonlinear wave equation $u_{tt}-Δu = u^p$ with $p\geq 7$ being an odd integer has a countable family of regular spherically symmetric self-similar solutions.
We give blow-up results for the Klein-Gordon equation and other perturbations of the semilinear wave equations with superlinear power nonlinearity, in one space dimension or in higher dimension under radial symmetry outside the origin.
We study a class of fractional semilinear elliptic equations and formulate the corresponding Calderón problem. We determine the nonlinearity from the exterior partial measurements of the Dirichlet-to-Neumann map by using first order linearization and the Runge approximation property.
Existence and non-existence results for a semilinear fractional Neumann problem
本文研究半线性非局部分数阶 Neumann 问题，通过建立非负解的先验 $L^\infty$ 估计证明非常数解的非存在性，并在超临界情形下得到径向非递减解的存在性，将已有分析扩展至 $0<s\le 1/2$。
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